**5. Results and discussion**

#### **5.1. Three-beam case**

Three-beam case is the most primitive case of X-ray multiple reflection. Figures 5[*E*(*a*)] and 5[*S*(*a*)] are 0 0 0-forward-diffracted and 4 0 4- and 0 4 4-reflected X-ray pinhole topograph images. Figures 5[*E*(*b*)] and 5[*S*(*b*)] are enlargements of 0 4 4-reflected images from Figures 5[*E*(*a*)] and 5[*S*(*a*)], respectively. Fine-fringe regions 1 and 2 ([*FFR*(1)] and [*FFR*(2)]) and 76 Recent Advances in Crystallography X-ray N-beam Takagi-Taupin Dynamical Theory and N-beam Pinhole Topographs Experimentally Obtained and Computer-Simulated <sup>11</sup> 77 X-Ray N-Beam Takagi-Taupin Dynamical Theory and N-Beam Pinhole Topographs Experimentally Obtained and Computer-Simulated

**Figure 5.** [*E*(*a*)] and [*S*(*a*)] (*x* ∈ {*a*, *b*}) are experimentally obtained and computer-simulated three-beam X-ray pinhole topographs with an incidence of horizontal-linearly polarized X-rays whose photon energy was 18.245 keV. [*Y*(*b*)] (*Y* ∈ {*E*, *S*}) are enlargements of 0 4 4-reflected X-ray images in [*Y*(*a*)]. The exposure time for [*E*(*x*)] was 600 s.

'Y-shaped' bright region (*YBR*) indicated by arrows in Figure 5[*S*(*b*)], are found also in Figure 5[*E*(*b*)].

#### **5.2. Four-beam case**

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Figure 4 is a reproduction of Fig 7 in reference [17] showing the experimental setup when the six-beam pinhole topographs shown in reference [17] were taken. Also in the case of *n*-beam pinhole topographs (*n* ∈ {3, 4, 5, 6, 8, 12}) the [1 1 1]-oriented floating-zone silicon crystal with thickness 9.6 mm (for three-, four-, five-, six- and eight-beam topographs) and 10.0 mm (for twelve-beam topographs) were also mounted on the four-axis goniometer whose *χ*-, *φ*-, *ω*and *θ*-axes can be rotated. Transmitted X-rays through the sample and two reflected X-rays were searched by three *PIN* photo diodes as shown in Figure 4. The positions of the two *PIN* photo diodes for detecting the reflected X-rays were determined using a laser beam guide reflected by a mirror. The mirror was mounted at the sample position on the goniometer whose angular positions were calculated such that the mirror reflects the laser beam to the

**Figure 4.** A schematic drawing of the goniometer on which the sample crystal was mount (reproduction

After adjusting the angular position of the goniometer such that the *n*-beam simultaneous reflection condition was satisfied, the size of slit *S* in Figure 3 (*a*) was limited to be 25×25 *μ*m. *N* images of *n*-beam pinhole topographs were simultaneously recorded on an imaging plate

Three-beam case is the most primitive case of X-ray multiple reflection. Figures 5[*E*(*a*)] and 5[*S*(*a*)] are 0 0 0-forward-diffracted and 4 0 4- and 0 4 4-reflected X-ray pinhole topograph images. Figures 5[*E*(*b*)] and 5[*S*(*b*)] are enlargements of 0 4 4-reflected images from Figures 5[*E*(*a*)] and 5[*S*(*a*)], respectively. Fine-fringe regions 1 and 2 ([*FFR*(1)] and [*FFR*(2)]) and

**4.2. Sample crystal**

of Figure 7 in reference [17]).

placed behind the sample crystal.

**5. Results and discussion**

**5.1. Three-beam case**

direction of X-rays to be reflected by the sample crystal.

Figures 6[*E*(*x*)] and 6[*S*(*x*)] (*x* ∈ {*a*, *b*, *c*}) show experimentally obtained and computer-simulated pinhole topographs of 0 0 0-forward-diffracted, and 0 6 6-, 6 2 8 and 6 2 4-reflected X-ray images, respectively. [*Y*(*a*)], [*Y*(*b*)] and [*Y*(*c*)] (*Y* ∈ {*E*, *S*}) were obtained with an incidence of +45◦-inclined-linearly, −45◦-inclined-linearly and right-screwed-circularly polarized X-rays, respectively, generated with the phase retarder system or assumed in the simulation.

Figures 7[*E*(*x*)] and 7[*S*(*x*)] (*x* ∈ {*a*, *b*, *c*}) are enlargements of 6 2 8-reflected X-ray images from Figures 6[*E*(*x*)] and 6[*S*(*x*)]. In Figure 7[*S*(*a*)], fine-fringe region 1 [*FFR*(1)], fine-fringe region 2 [*FFR*(2)] and knife-edge line (*KEL*) are indicated by arrows. These characteristic patterns are also observed in Figure 7[*E*(*a*)]. In Figures 7[*E*(*b*)] and 7[*S*(*b*)], while *FFR*(2) is

**Figure 6.** [*E*(*x*)] and [*S*(*x*)] (*x* ∈ {*a*, *b*, *c*}) are experimentally obtained and computer-simulated four-beam X-ray pinhole topographs with an incidence of +45◦-inclined-linearly, −45◦-inclined-linearly and right-screwed-circularly polarized X-rays whose photon energy was 18.245 keV. The exposure time for [*E*(*x*)] was 1800 s.

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**Figure 6.** [*E*(*x*)] and [*S*(*x*)] (*x* ∈ {*a*, *b*, *c*}) are experimentally obtained and computer-simulated

for [*E*(*x*)] was 1800 s.

four-beam X-ray pinhole topographs with an incidence of +45◦-inclined-linearly, −45◦-inclined-linearly and right-screwed-circularly polarized X-rays whose photon energy was 18.245 keV. The exposure time

**Figure 7.** [*E*(*x*)] and [*S*(*x*)] (*x* ∈ {*a*, *b*, *c*}) are enlargements of 6 2 8-reflected X-ray images in Figures 6 [*E*(*x*)] and 6 [*S*(*x*)].

observed at the same position, a pattern like a fish born (*PFB*) is observed in place of [*FFR*(1)]. *KEL* in Figures 7[*E*(*b*)] and 7[*S*(*b*)] are fainter. Furthermore, an arched line (*AL*) and a bright region (*BR*) not observed in Figures 7[*E*(*a*)] and 7[*S*(*a*)] are observed in Figure 7[*E*(*b*)] and 7[*S*(*b*)]. In Figures 7[*E*(*c*)] and 7[*S*(*c*)], almost all the characteristic patterns above-mentioned are observed.

Between the horizontal and vertical components of incident X-rays, there is difference not in amplitude but in phase among Figures [*Y*(a)], [*Y*(b)] and [*Y*(c)] (*Y* ∈ {*E*, *S*}), which reveals that the wave fields excited by horizontal- and vertical-linearly polarized components of the incident X-rays interfere with each other.

## **5.3. Five-beam case**

**Figure 8.** [*E*(*x*)] and [*S*(*x*)] (*x* ∈ {*a*, *b*}) are experimentally obtained and computer-simulated five-beam X-ray pinhole topographs with an incidence of vertical-linearly polarized X-rays whose photon energy was 18.245 keV. [*Y*(*b*)] (*Y* ∈ {*E*, *S*}) are enlargements of 5 5 5-reflected X-ray images in [*Y*(*a*)]. The exposure time for [*E*(*x*)] was 1800 s.

In the case of cubic crystals, five reciprocal lattice nodes (including the origin of reciprocal space) can ride on a circle in reciprocal space. For understanding such a situation, refer to Figure 1 of reference [17].

Figures 8[*E*(*a*)] and 8[*S*(*a*)] are experimentally obtained and computer-simulated five-beam pinhole topographs. Figures 8[*E*(*b*)] and 8[*S*(*b*)] are enlargements of 5 5 5-reflected X-ray images from Figures 8[*E*(*a*)] and 8[*S*(*a*)]. Knife-edge patterns 1 and 2 [*KEL*(1) and *KEL*(2)] and 'harp-shaped' pattern (*HpSP*) indicated by arrows in Figure 8[*S*(*b*)] are observed also in Figure 8[*E*(*b*)].

Remarking on the directions of *KEL*(1) and *KEL*(2), these knife-edge patterns are directed to 0 0 0-forward-diffracted and 3 3 3-reflected X-ray images, respectively. Then, *KEL*(1) and *KEL*(2) are considered to suggest the strong energy exchange mechanism between 0 0 0-forward-diffracted and 5 5 5-reflected X-ray wave fields and between 3 3 3- and 5 5 5-reflected X-ray wave fields. Such knife-edge patterns are found also in three-, four-, six- and eight-beam pinhole topograph images shown in the present chapter.

#### **5.4. Six-beam case**

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observed at the same position, a pattern like a fish born (*PFB*) is observed in place of [*FFR*(1)]. *KEL* in Figures 7[*E*(*b*)] and 7[*S*(*b*)] are fainter. Furthermore, an arched line (*AL*) and a bright region (*BR*) not observed in Figures 7[*E*(*a*)] and 7[*S*(*a*)] are observed in Figure 7[*E*(*b*)] and 7[*S*(*b*)]. In Figures 7[*E*(*c*)] and 7[*S*(*c*)], almost all the characteristic patterns above-mentioned

Between the horizontal and vertical components of incident X-rays, there is difference not in amplitude but in phase among Figures [*Y*(a)], [*Y*(b)] and [*Y*(c)] (*Y* ∈ {*E*, *S*}), which reveals that the wave fields excited by horizontal- and vertical-linearly polarized components of the

**Figure 8.** [*E*(*x*)] and [*S*(*x*)] (*x* ∈ {*a*, *b*}) are experimentally obtained and computer-simulated five-beam X-ray pinhole topographs with an incidence of vertical-linearly polarized X-rays whose photon energy was 18.245 keV. [*Y*(*b*)] (*Y* ∈ {*E*, *S*}) are enlargements of 5 5 5-reflected X-ray images in [*Y*(*a*)]. The

In the case of cubic crystals, five reciprocal lattice nodes (including the origin of reciprocal space) can ride on a circle in reciprocal space. For understanding such a situation, refer to

are observed.

**5.3. Five-beam case**

exposure time for [*E*(*x*)] was 1800 s.

Figure 1 of reference [17].

incident X-rays interfere with each other.

**Figure 9.** [*E*(*a*)] and [*S*(*a*)] are experimentally obtained and computer-simulated six-beam X-ray pinhole topographs with an incidence of horizontal-linearly polarized X-rays with a photon energy of 18.245 keV. [*E*(*b*)] and [*S*(*b*)] are enlargements of 2 6 4- and 0 6 6-reflected X-ray images in [*E*(*a*)] and [*S*(*a*)]. The exposure time for [*E*(*a*)] and [*E*(*b*)] was 300 s.

While experimental and computer-simulated six-beam pinhole topograph images whose shapes are regular hexagons have been reported in reference [14, 16, 17], shown in this

#### 16 Will-be-set-by-IN-TECH 82 Recent Advances in Crystallography

section are six-beam pinhole topographs whose Borrmann pyramid is not a regular hexagonal pyramid.

Such six-beam pinhole topographs experimentally obtained and computer-simulated are shown in Figure 9. Figures 9[*E*(*b*)] and 9[*S*(*b*)] are enlargements of 2 6 4- and 0 6 6-reflected X-ray images from Figures 9[*E*(*a*)] and 9[*S*(*a*)]. Knife-edge patterns [*KEL*(1) and *KEL*(2)] indicated by arrows in Figure 9[*S*(*b*)] are found also in Figure 9[*E*(*b*)]. Circular patterns that were found in the central part of the six-beam pinhole topographs [14, 16, 17] cannot be found in the present case. A 'heart-shaped' pattern (*HSP*) is found also in Figure 9[*E*(*b*)].
